AMC12 2012 A

AMC12 2012 A · Q9

AMC12 2012 A · Q9. It mainly tests Divisibility & factors, Remainders & modular arithmetic.

A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?

一年是闰年当且仅当该年份能被 400 整除（如 2000 年），或能被 4 整除但不能被 100 整除（如 2012 年）。小说家查尔斯·狄更斯诞辰 200 周年纪念日是 2012 年 2 月 7 日（星期二）。狄更斯出生那天是星期几？

(A) Friday 星期五

(B) Saturday 星期六

(D) Monday 星期一

(E) Tuesday 星期二

Answer

Correct choice: (A)

正确答案：（A）

Solution

In this solution we refer to moving to the left as decreasing the year or date number and moving to the right as increasing the year or date number. Every non-leap year we move to the right results in moving one day to the right because $365\equiv 1\pmod 7$. Every leap year we move to the right results in moving $2$ days to the right since $366\equiv 2\pmod 7$. A leap year is usually every four years, so 200 years would have $\frac{200}{4}$ = $50$ leap years, but the problem says that 1900 does not count as a leap year. Therefore there would be 151 regular years and 49 leap years, so $1(151)+2(49)$ = $249$ days back. Since $249 \equiv 4\ (\text{mod}\ 7)$, four days back from Tuesday would be $\boxed{\textbf{(A)}\ \text{Friday}}$.

在本解答中，我们把向左移动理解为年份或日期数减小，向右移动理解为年份或日期数增大。每向右跨过一个非闰年，星期向右移动一天，因为 $365\equiv 1\pmod 7$。每向右跨过一个闰年，星期向右移动 $2$ 天，因为 $366\equiv 2\pmod 7$。闰年通常每四年一次，所以 200 年应有 $\frac{200}{4}$ = $50$ 个闰年，但题目说明 1900 年不算闰年。因此共有 151 个平年和 49 个闰年，所以向左回退的天数为 $1(151)+2(49)$ = $249$ 天。由于 $249 \equiv 4\ (\text{mod}\ 7)$，从星期二向左回退四天是 $\boxed{\textbf{(A)}\ \text{Friday}}$。

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